OFFSET
0,3
FORMULA
a(n) = n! * Sum_{k=0..floor(n/2)} Stirling2(n-k,k)/(n-2*k+1)!.
From Vaclav Kotesovec, Mar 11 2024: (Start)
E.g.f.: 1 - x - LambertW(-exp((1 - x)*x)*x^2)/x.
a(n) ~ sqrt(2 + r - 2*r^2) * n^(n-1) / (exp(n) * r^(n+1)), where r = 0.5356007344755967412570670018666980389185523835846... if the root of the equation exp(1 + r - r^2) * r^2 = 1. (End)
MATHEMATICA
nmax = 20; CoefficientList[Series[1 - x - ProductLog[-E^((1 - x)*x)*x^2]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 11 2024 *)
PROG
(PARI) a(n) = n!*sum(k=0, n\2, stirling(n-k, k, 2)/(n-2*k+1)!);
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Mar 11 2024
STATUS
approved